A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

TL;DR

This paper introduces a comprehensive framework for constructing conservative numerical methods for nonlinear dispersive wave equations, ensuring the conservation of key invariants while employing explicit relaxation Runge-Kutta time integration.

Contribution

The paper presents a novel, unified approach combining summation by parts operators, split forms, and relaxation Runge-Kutta methods to develop fully-discrete conservative schemes for various dispersive wave equations.

Findings

The methods conserve all linear invariants and one nonlinear invariant for each system.

Numerical tests demonstrate the schemes' favorable stability and accuracy.

Applicable to finite difference, spectral, and finite element spatial discretizations.

Abstract

We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests.